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Unit 3, Lesson 4

The Shape of Distributions

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Integrated Math 1

4.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Which Three Go Together?

Materials

None

Activity Narrative

This Warm-up prompts students to compare four distributions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Before students begin, consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. Signals might include a thumbs-up or a certain number of fingers that tells the number of responses they have. Using such subtle signals is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.

Launch

Arrange students in groups of 2–4. Display the dot plots for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three dot plots that go together and can explain why. Next, tell students to share their response with their group, and then together to find as many sets of three as they can.

Activity

None

Which three go together? Why do they go together?

Dot plot from negative 21 to negative 9 by 1’s. Beginning at negative 21, number of dots above each increment is 0, 1, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0. — A.

Dot plot from negative 21 to negative 9 by 1's. Beginning at negative 21, number of dots above each increment is 0, 6, 4, 3, 2, 1, 1, 2, 3, 4, 6, 0, 0. — B.

Dot plot from negative 21 to negative 8 by 1’s. Beginning at negative 21, number of dots above each increment is 0, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0. — C.

Dot plot from negative 21 to negative 9 by 1’s. Beginning at negative 21, number of dots above each increment is 0, 1, 1, 1, 1, 2, 3, 4, 6, 7, 4, 3, 0. — D.

Activity Synthesis

Invite each group to share one reason why a particular set of three goes together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.

During the discussion, prompt students to explain the meaning of any statistical terminology they use, such as “symmetric,” “skewed,” “uniform,” “bimodal,” and “bell-shaped,” and to clarify their reasoning as needed. Consider asking:

“How do you know . . . ?”
“What do you mean by . . . ?”
“Can you say that in another way?”

4.2

Activity

Instructional Routines

Card Sort MLR7: Compare and Connect Take Turns

Materials

To Copy (from Blackline Masters)

Matching Distributions Cards

Activity Narrative

The mathematical purpose of this activity is to give students a chance to practice finding data displays that represent the distribution of the same data set and to practice using precise vocabulary for describing the shape of the distributions. In this partner activity, students take turns matching displays of distributions. As students trade roles, they explain their thinking and listen to the reasoning of a partner. This provides students with opportunities both to explain their reasoning and to critique the reasoning of others (MP3).

Students sort different distributions during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Monitor for different ways groups choose to categorize the distributions, but especially for categories that distinguish between the shapes such as symmetric, bell-shaped, or uniform distributions.

4.3

Activity

Instructional Routines

MLR2: Collect and Display

Materials

None

Activity Narrative

The mathematical purpose of this activity is to remind students of the importance of context to statistics. Although some analysis can be done outside of a context, it is often useful to think about the real situations in which the data were collected, to engage student intuition and understanding.

Launch

Keep students in the same groups. Assign each pair of students one of the completed matches from the Card Sort activity. Tell students that there are many possible answers for each representation. After 2 minutes of quiet work time, ask students to compare their responses to their partner’s and decide if they are both reasonable. You may need to demonstrate this activity before beginning if you think students may have trouble getting started. After each group finishes with their assigned distribution, assign the group another distribution to consider.

MLR2 Collect and Display. Circulate, listen for, and collect the language that students use as they share their educated guesses of the survey question that produced the data for their assigned matches. On a visible display, record collected words and phrases, such as “line of symmetry,” “bell-shaped,” “skewed,” “bimodal,” or “uniform.” Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading

Action and Expression: Develop Expression and Communication. To help get students started, display sentence frames, such as “A possible survey question that produced this data is _____ because _____.”
Supports accessibility for: Language, Organization

Lesson Synthesis

In this lesson students describe the shape of distributions using formal language and invent contexts for distributions with different shapes. Here are some questions for discussion.

“What does a symmetric data set look like?” (It will have a line of symmetry in the middle and the left side will look like a reflection of the right side.)
“What does it mean to say that the shape of a distribution is uniform?” (There will be an equal number of each data value and the shape will look rectangular.)
“Have you heard of a bell curve before? How does this relate to a bell-shaped distribution?” (Yes. I have heard of it in science class where a bell curve was used to compare data in an experiment.)
“What is an example of a context where you would expect to find a bimodal distribution?” (You might find it if you measured the weight of a herd of cows in the springtime. The adult cows would be one peak and the calves would be the other peak.)
“Can a skewed distribution also be symmetric? Why or why not?” (No, because skewed means that one side of the peak of the data has more data values further away from the peak than the other side. There is no line of symmetry.)

to access Cool-down.

Student Lesson Summary

We can describe the shape of distributions as symmetric, skewed, bell-shaped, bimodal, or uniform. The dot plot, histogram, and box plot shown here represent the distribution of the same data set. This data set has a symmetric distribution.

<p>Dot plot from 6 to 18 by 1’s. Beginning at 6, number of dots above each increment is 0, 1, 1, 2, 3, 5, 8, 5, 3, 2, 1, 1, 0.</p>

<p>Histogram from 6 to 18 by 2’s. Beginning at 6 up to but not including 8, height of bar at each interval is 1, 3, 8, 10, 5, 2</p>

<p>Box plot from 6 to 18 by 1’s. Whisker from 7 to 11. Box from 11 to 13 with vertical line at 12. Whisker from 13 to 17.</p>

In a symmetric distribution, the mean is equal to the median and there is a vertical line of symmetry in the center of the data display. The histogram and the box plot both group data together. Since histograms and box plots do not display each data value individually, they do not provide information about the shape of the distribution to the same level of detail that a dot plot does. This distribution, in particular, can also be called bell-shaped. A bell-shaped distribution has a dot plot that takes the form of a bell with most of the data clustered near the center and fewer points farther from the center. This makes the measure of center a very good description of the data as a whole. Bell-shaped distributions are always symmetric or close to it.

The dot plot, histogram, and box plot shown here represent a skewed distribution.

<p>Dot plot from 8 to 18 by 1’s. Beginning at 8, number of dots above each increment is 6, 5, 4, 3, 2, 2, 1, 1, 1, 0, 0.</p>

<p>Histogram from 8 to 18 by 2’s. Beginning at 8 up to but not including 10, height of bar at each interval is 11, 7, 4, 2, 1.</p>

<p>Box plot from 8 to 18 by 1’s. Whisker from 8 to 8.5. Box from 8.5 to 12 with vertical line at 10. Whisker from 12 to 16.</p>

In a skewed distribution, one side of the distribution has more values farther from the bulk of the data than the other side has. This results in the mean and median not being equal. In this skewed distribution, the data are skewed to the right because most of the data are near the 8 to 10 interval, but there are many points to the right. The mean is greater than the median. The large data values to the right cause the mean to shift in that direction while the median remains with the bulk of the data. So, the mean is greater than the median for distributions that are skewed to the right. In a data set that is skewed to the left, a similar effect happens but to the other side. Again, the dot plot provides a greater level of detail about the shape of the distribution than do either the histogram or the box plot.

A uniform distribution has the data values evenly distributed throughout the range of the data. This causes the distribution to look like a rectangle.

<p>Dot plot from 7 to 19 by 1’s. Beginning at 7, number of dots above each increment is 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0.</p>

<p>Histogram from 7 to 19 by 2’s. Beginning at 7 up to but not including 9, height of bar at each interval is 4.</p>

<p>Box plot from 7 to 19 by 1’s. Whisker from 7 to 9.5. Box from 9.5 to 15.5 with vertical line at 12.5. Whisker from 15.5 to 18.</p>

In a uniform distribution, the mean is equal to the median because a uniform distribution is also a symmetric distribution. The box plot does not provide enough information to describe the shape of the distribution as uniform, though the even length of each quarter does suggest that the distribution may be approximately symmetric.

A bimodal distribution has two very common data values seen in a dot plot or histogram as distinct peaks.

<p>Dot plot from 6 to 20 by 1’s. Beginning at 6, number of dots above each increment is 0, 5, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 4, 1, 0.</p>

<p>Histogram from 6 to 20 by 2’s. Beginning at 6 up to but not including 8, height of bar at each interval is 5, 4, 2, 2, 2, 4, 5.</p>

<p>Box plot from 6 to 20 by 1’s. Whisker from 7 to 8. Box from 8 to 17 with vertical line at 12.5. Whisker from 17 to 19.</p>

Sometimes, a bimodal distribution has most of the data clustered in the middle of the distribution. In these cases, the center of the distribution does not describe the data very well. Bimodal distributions are not always symmetric. For example, the peaks may not be equally spaced from the middle of the distribution, or other data values may disrupt the symmetry.

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Lesson 4

The Shape of Distributions

Warm-up

Standards Alignment

Building On

Addressing

HSS-ID.A.1

Building Toward

HSS-ID.A.2

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSS-ID.A.1

Building Toward

HSS-ID.A.2

Standards Alignment

Building On

Addressing

HSS-ID.A.1

Building Toward

HSS-ID.A.2